If I own a trademark does that entitle me to the rights of the domain name?

That is correct, you don't need a trademark to get a domain name, however people have to be really careful with trademarks. For instance Ian Tunes who registered "iTunes" before Apple's "itunes.co.uk" existed. He could keep the domain name for life, as he was the first to register it, but because Apple was making offer after offer and giving him a lot of hassle to get this name, one day he was fed up and redirected his domain itunes.co.uk to Napster (biggest iTunes competitor at that time). From that point, he breached Apple's iTunes copyright and lost the domain name (Instead of making a million by accepting the original deal).